Talk:Monomial

old comments

What contradiction(s?) as implied by the "may not be consistent" arises due to the allowance of coefficients?

The monomials in one unknown X are therefore the powers Xⁿ for n = 0, 1, 2,...

Doesn't that "0" imply that coefficients are permitted?

It implies that 1 is a monomial.

Charles Matthews 14:58, 5 Nov 2003 (UTC)

Right, right, I was just coming back to remove my dumb comment :)

'Diverge' - an English subjunctive, no?

Charles Matthews 15:20, 5 Nov 2003 (UTC)

Oh. Sorry. Didn't realize it was subjunctive.

Definition: what kind of powers?

"is a product of powers of variables". Do these have to be nonzero integral powers? Is x√x a monomial? What about 1/x ?Gphilip (talk) 08:15, 3 December 2012 (UTC)[reply]

The first sentence says "in the context of polynomials". Thus, implicitly, non negative integer exponents are assumed. I agree that it would be better to make this assumption explicit. On the other hand monomials with negative exponents are considered in the context of Laurent polynomials and monomials with rational numbers as exponents are considered in the context pf Puiseux series. I'll edit the lead accordingly. D.Lazard (talk) 09:59, 3 December 2012 (UTC)[reply]

Recent reverts

The enemies of god‎ has done a major revision of the article. This revision includes

Removal of essential information, namely the existence in the literature of two different, although related, definitions of a monomial.
Introduction in the body of the article of external links which should be removed per WP:ELNO, even if they were in a section "External links". Also introduction of a link to a dab page
Introduction of contradiction between two consecutive paragraph: In a paragraph is is said that 6 and $-7x^{5}$ are monomials, and, the next paragraph asserts that, in the case of one variable, a monomial is either 1 or a power of the variable with a positive integer as an exponent (solving this contradiction requires considering the two different definitions, whose description has been removed by the editor)
Introduction of a terminology that is uncommon in mathematics, and therefore confusing ("lone variable")
Introduction of a terminology ("atomic term") that is not defined, nor linked, irrelevant here, and too WP:TECHNICAL for this article (this is a term of mathematical logic).
Confusing use of two different meanings of "term" in the same sentence: "atomic term" refers to terms of a theory in logic, and later in the sentence, "term" is used as a synonymous of "summand".

This is not a complete list of the issues of this edit, but is sufficient to show that it makes the article much worst than it was. Thus this edit cannot be accepted.

I have reverted this edit twice. I have also opened a thread in the talk page of the editor, and warned him of the WP:3RR rule. In each case, "the enemies" has restored his edit, without any comment here and in talk pages (here, mine, or his).

Thus, as this editor seems not accept discussing, I'll revert him again, and if he continue to edit warring, I'll notify this behavior to Administrators' noticeboard. D.Lazard (talk) 19:33, 17 January 2019 (UTC)[reply]

I concur with D. Lazard that the new version appears to be inferior to the old one (thus, the reverting is warranted) For example, the new version had “a monomial is a simple product” as the first sentence; this is not understandable and remember like the overwhelming majority of the readers read only the first sentence. But I would like also to say that I’m not too fond of the lead of the stable version; it’s overly technical. We shouldn’t be showering a calculus student with the term like “power product”. Of course, the precision is necessary in the beyond-calculus levels; foe example, one may need to distinguish between

xy

and

yx

but that can be done later in the article where we lose calculus students. —- Taku (talk) 18:48, 18 January 2019 (UTC)[reply]

There is a definition of monomial that I really like: it is a function from a finite set X to

\mathbb {N}

; see for example [1]. If

X=\{x_{1},\dots ,x_{n}\}

, then it’s easy see this definition is equivalent to the usual one but has a benefit that we are not implicitly introducing an ordering (which can be a delicate structure). (If I recall, there is also a definition in terms of an orbit.) —- Taku (talk) 18:59, 18 January 2019 (UTC)[reply]

I am also not too fond with the stable version. The definition proposed by Taku is very elegant, and deserve to be mentioned in the article. In fact, monomials may be identified with multisets, and this identification is often used for counting multisets and monomials. However, this should not appear as the main definition, as it is confusing, because it hides the main property of monomials, that is to be specific polynomials that form a basis of the vector space or module of all polynomials. D.Lazard (talk) 20:35, 18 January 2019 (UTC)[reply]

No disagreement from me; after all, it’s “monomial” as opposed to “polynomial”. —- Taku (talk) 01:37, 20 January 2019 (UTC)[reply]

Degree of a monomial

I'm a bit surprised to find that this page asserts the degree of a monomial to be "the sum of all the exponents of the variables". This article even references CLO without mentioning that there are other possible definitions of "degree" for multivariate monomials. That is, this article is written entirely in the frame of total degree. I do agree that this is the most common notion of a degree in multivariate contexts, but it feels amiss to not point out the article's conventional choice.

Additionally, this definition is only codified after an extensive discussion of the number of monomials of a certain degree. --EuclideanSwag (talk) 09:06, 22 November 2022 (UTC)[reply]

Figure recently added

An editor has recently added an image to the article, and has added it again after a revert. I'll revert this image again, because of multiples issues. Here are the main ones.

MOS:PERTINENCE: There is nothing in the body of the article that is illustrated by this image
Unusual terminology that is not defined in Wikipedia: the reader has to guess himself that "end behaviour" means "behavior at infinity or asymptotic behavior.
The image is about univariate polynomial functions defined by a monomial, and is placed in a section that is explicitely about multivariate monomials.
There is nothing specific about monomials in this image: all univariate polynomial functions of degree $n$ have the same behavior at infinity as their monomial of highest degree.
The image contains textual information that cannot be easily edited.
The textual information consists of formulas without any prose for explaining them
The formulas are awfully formatted and use uncommon notation not defined in Wikipedia: the variables are not in italic; "n%2=0" instead of " $n mod 2 = 0$ " or, better, " $n$ even" ("%" means "mod" in some programming languages, but not in mathematics); notation "x → +∞, f(x) → +∞" is nowhere defined in Wikipedia; apparently it means ${\textstyle \lim _{x\to +\infty }f(x)=+\infty .}$
etc.

D.Lazard (talk) 21:52, 12 December 2022 (UTC)[reply]